Problem 7.3 — Compute L(Modg) explicitly for small g≥ 2.

Definitions and mathematical statement

Let $\lambda(f)>1$ be the stretch factor of a pseudo-Anosov mapping class $f\in\operatorname{Mod}_g$, characterized by scaling of its invariant measured foliations. Write \[D_g=\inf\{\lambda(f):f\in\operatorname{Mod}_g\text{ is pseudo-Anosov}\}.\] The problem asks to compute the source's $L(\operatorname{Mod}_g)$ explicitly for small $g\geq2$. Its discussion of algebraic degree and minimal dilatation suggests $L(\operatorname{Mod}_g)=D_g$, but the defining preceding notation is absent. If $L$ instead denotes logarithmic dilatation/Teichmüller translation length, the target would be $\log D_g$; these cannot silently be identified. No restriction to orientable invariant foliations or a special mapping-class subgroup is stated.

An explicit exact computation should identify the least value in the intended convention, exhibit a pseudo-Anosov attaining it, and justify exclusion of every smaller value. A numerical approximation or upper-bound example alone does not specify the minimum.

The source describes an in-principle procedure for a fixed $g$: bound the algebraic degrees of possible dilatations, obtain bounds/candidates from algebraic integers in that degree range, enumerate train-track descriptions and pseudo-Anosov candidates, bound when dilatations become large, and match the two lists to certify a minimum. It calls practical execution difficult even for small genus. Its notation $\lambda(\alpha)$ is described as the "largest root" of an algebraic integer's minimal polynomial; this needs a convention for complex roots (largest positive real root versus maximal modulus) and admissible Perron-type candidates. The requisite finite enumeration, stopping bounds and normalization are not supplied, so this outline is retained as a proposed method rather than a validated algorithm.

The research summary reports some small-genus progress, especially genus two and three, but mixes golden/silver-ratio descriptions, a tentative $(3+\sqrt5)/2$ value, and qualified resolution claims without usable references. Those conflicting numerical claims are not inserted as the value of $D_g$ or $L$.

Short English statement

Determine exact least pseudo-Anosov dilatations in small genera, with a realizing example and a matching lower bound. The source's $L$ normalization and the inconsistent small-genus claims in its research summary require verification.

Sources

• Local UnsolvedMath registry, permanent ID $11000079$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 7.3, PDF page 57: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• The research summary mentions Zhang and Cho and computer-search work without precise citations; its proposed values and attribution are unverified and internally qualified.